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For D(ft(x)) is the limit, as h ^ 0 tends to 0, of the mapping

and the result therefore follows from (12.16.5).

The mapping f'to is said to be the derivative at the point t0 of the mapping
tt-+ft (with respect to the topology of simple convergence), or the weak
derivative when F is the field of scalars (and therefore &(E; F) = E').


( Let A be an open set in C and let zi-/z be a weakly analytic
mapping (12.15) of A into the dual E' of a complex Frechet space F. Then the
same argument as in (12.16.6) shows that there exists a weak derivative zt-+fz
of z\-*fz9 with fz(x) = D(fz(x)) for all xe E, and this weak derivative is
weakly analytic. Moreover, for each a e A and each circuit y contained in
A  {a}, we have the Cauchy formula

(                   ;(* ;?)/(*)

= J^ {fz(x)dz

2nijy   z-a

Conversely, let y be a road in C defined in an interval I = \b, c] in R, and
let z\-+gz be a weakly continuous mapping of y(I) into E'. Then, for z
the mapping


f 9-

Jy     C

is a linear form/z on E, belonging to E'. Indeed, for each x e E the right-hand
side of ( is the limit of a sequence

n jt=o  U - z


((3.16.5) and (8.7.8)), hence our assertion is a consequence of the Banach-
Steinhaus theorem. By virtue of (9.9.2), fz is weakly analytic in C  y(I), and
for a $ y(I) we can write

(                       fjx) = f CB(X)(Z - a)",

n = 0

the series being convergent in any disk with center a not meeting y(I). The of extension of identities.